Your problem is \hline\\ which could be
\hline &&& \\[-1.5ex]
(experiment the value). In general it's best to use ex units for vertical spacing.
I propose three realizations of the table, in decreasing order of ugliness.
\documentclass{article}
\usepackage[T1]{fontenc}
\usepackage[italian]{babel}
\usepackage{amsmath}
\usepackage{array}
\usepackage{booktabs}
\begin{document}
\begin{center}
\begin{tabular}{
w{c}{0.22\textwidth}|
w{c}{0.20\textwidth}|
w{c}{0.20\textwidth}|
w{c}{0.20\textwidth}
}
& $x=10$ & $x=100$ & $x=1000$ \
\hline
&&&\[-1.5ex]
$y=\log_{10}x$ & $1$ & $2$ & $3$ \[1ex]
$y=x^2$ & $10^2$ & $10^4$ & $10^6$ \[1ex]
$y=10^x$ & $10^{10}$ & $10^{100}$ & $10^{1000}$
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{\textwidth}{@{\extracolsep{\fill}}cccc@{}}
\toprule
& $x=10$ & $x=100$ & $x=1000$ \
\midrule
$y=\log_{10}x$ & $1$ & $2$ & $3$ \
\addlinespace
$y=x^2$ & $10^2$ & $10^4$ & $10^6$ \
\addlinespace
$y=10^x$ & $10^{10}$ & $10^{100}$ & $10^{1000}$ \
\bottomrule
\end{tabular}
\end{center}
\begin{center}
\setlength{\tabcolsep}{1em}
\begin{tabular}{@{}lccc@{}}
\toprule
Funzione & \multicolumn{3}{c@{}}{Valore di $x$} \
\cmidrule{2-4}
& $10$ & $100$ & $1000$ \
\midrule
$y=\log_{10}x$ & $1$ & $2$ & $3$ \
\addlinespace
$y=x^2$ & $10^2$ & $10^4$ & $10^6$ \
\addlinespace
$y=10^x$ & $10^{10}$ & $10^{100}$ & $10^{1000}$ \
\bottomrule
\end{tabular}
\end{center}
\end{document}
Avoid [H]; if there is no caption, a center environment suffices. If there's a caption, let the environment float.

I left “y=”, but I consider it bad mathematical usage.
\\after\hline– David Carlisle Oct 11 '21 at 16:40